Subtraction of Polynomials

Subtracting polynomials involves adding the additive inverse of the second polynomial to the first.

It requires changing the signs of the terms of the polynomial being subtracted and then combining the like terms.

There are three components of a polynomial:

Coefficients: These are constant values like -1, 4, etc.

Variables: These are unknown quantities like x, y, z, etc.

Operators: For subtraction, the operator is the minus (-) symbol.

When subtracting polynomials, students should follow the list of rules:

Flip signs: Always flip the signs of each term of the second polynomial and perform addition.

Combine like terms: Only like terms can be subtracted from one another, so group all like terms together.

Simplifying result: After all like terms are combined, there will be unlike terms remaining.

Write the remaining unlike terms as they are, along with the like terms, to get the final result.

The following are the methods of subtraction of polynomials:

Method 1: Horizontal Method

To apply the horizontal method for subtraction of polynomials, use the following steps.

Step 1: Write both polynomials in the same line using a minus sign in between.

Step 2: Remove the brackets and change the signs of the second polynomial.

Step 3: Combine the like terms.

Let’s apply these steps to an example:

Question: Subtract

Step 1: Write both polynomials in the same line,

Step 2: Remove the brackets and change the signs of the second polynomial 5x and 2x are like terms having the same variable x, similarly, 3y and -y are also like terms.

Step 3: Write like terms together:

Answer:

Method 2: Column Method

When subtracting the polynomials using the column method, we write the polynomials one below the other. Make sure like terms are aligned in each column. Then change the signs of the second polynomial and add the polynomials.

For example, Subtract

Solution: Arrange the like terms vertically in columns 5x + 3y - 2 ← Minuend (from which we subtract) - 2x - y + 4 ← Subtrahend (what we subtract) ----------------------- 3x + 4y - 6 Therefore, upon subtracting, we get 3x + 4y - 6

In algebra, subtraction has some characteristic properties.

These properties are listed below:

Subtraction is not commutative

In subtraction, changing the order of the terms changes the result, i.e., A - B ≠ B - A

Subtraction is not associative Unlike addition, we cannot regroup in subtraction.

When three or more expressions are involved, changing the grouping changes the result.

(A − B) − C ≠ A − (B − C)

Subtraction is the addition of the opposite sign Subtracting a polynomial is the same as adding its opposite, so to make calculations easier, you can convert subtraction into addition by changing the signs of the second term.

A − B = A + (−B)

Subtracting zero from a polynomial leaves the polynomial as is Subtracting zero from any polynomial results in the same polynomial: A - 0 = A

Tips and tricks are useful for students to efficiently deal with the subtraction of polynomials. Some helpful tips are listed below:

Tip 1: Always pay attention to signs before combining like terms.

Tip 2: If two polynomials have identical terms, cross them out before starting the subtraction. This makes the polynomials shorter and provides more clarity due to fewer terms.

Tip 3: Beginners and visual learners can benefit from using the box model or column method to avoid missing signs and mismatching terms.

Subtraction in algebra is comparatively more challenging than addition, often leading to common mistakes. However, being aware of these errors can help students avoid them.